Dynamical system. The set of functions (signals) w : T W from T to W is denoted by W T. W variable space. T R time axis. W T trajectory space

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1 Dynamical system The set of functions (signals) w : T W from T to W is denoted by W T. W variable space T R time axis W T trajectory space A dynamical system B W T is a set of trajectories (a behaviour). w B means that w is a possible trajectory of the system B Note: the set definition is extremely general (and therefore abstarct). For example, it is not specialized to linear time-invariant systems. ELEC 3035 (Part I, Lecture 2) State space and polynomial representations 4 / 19

2 Representations of dynamical systems Systems are often described by equations f (w) = 0, f : W T R g, via representations B = {w W T f (w) = 0}. (repr) Note: f (w) = 0 is a specific but nonunique description of B. We will consider systems, which variable space is R w and time axis T = R continuous-time systems, or T = Z discrete-time systems. ELEC 3035 (Part I, Lecture 2) State space and polynomial representations 5 / 19

3 Linear time-invariant systems Properties of a system are defined in terms of its behaviour B and are translated to equivalent statements in terms of representations. B is linear if w,v B = αw + βv B, for all α,β R Recall the shift operator (σw)(t) = w(t + 1). B is time-invariant if w B = σ t w B, for all t. ELEC 3035 (Part I, Lecture 2) State space and polynomial representations 6 / 19

4 Input/output (I/O) partitioning Let Π R w w be a permutation matrix, and define u := Πw (I/O) y (This is just a reordering of the variables.) The variable u is an input if the behaviour associate with u if free, i.e., B u := {u (R m ) T there is y such that Π 1 u B } = (R m ) T. y (I/O) is an I/O partitioning for B if u is free and dim(u) is maximal. We will consider systems with given I/O partition and w.l.g. assume that Π = I. ELEC 3035 (Part I, Lecture 2) State space and polynomial representations 7 / 19

5 The difference equation Difference equations R 0 w(t) + R 1 w(t + 1) + + R l w(t + l) = 0, for all t Z is more compactly written using the shift operator σ as R 0 σ 0 w + R 1 σ 1 w + + R l σ l w = 0. ( ) Define the polynomial matrix R(z) = R 0 + R 1 z + + R l z l R g w [z] and note that R(σ)w = 0 is a convenient short hand notation for ( ). ELEC 3035 (Part I, Lecture 2) State space and polynomial representations 8 / 19

6 The differential equation Differential equations R 0 d 0 dt 0 w + R 1 is more compactly written as R d 1 dt 1 w + + R l d dt w = 0, where again R is the polynomial matrix d l dt l w = 0 R(z) = R 0 + R 1 z + + R l z l. For continuous-time systems, redefine σ as the derivative operator d/dt, so R(σ)w = 0 is a difference/differential eqn., depending on the context. ELEC 3035 (Part I, Lecture 2) State space and polynomial representations 9 / 19

7 Input/output representation The difference (in discrete-time) or differential (in continuous-time) eqn P(σ)y = Q(σ)u, P R g p [z], Q R g m [z] (I/O eqn) defines an LTI system B via B i/o (P,Q) := {w = (u,y) (R w ) N (I/O eqn) holds} (I/O repr) If g = p and det(p) = 0, (I/O repr) is called an input/output repr. The class of system that admit (I/O repr) is called finite dimensional. ELEC 3035 (Part I, Lecture 2) State space and polynomial representations 10 / 19

8 Transfer function Consider a system B i/o (P,Q) and let L be the Laplace transform. P( d dt )y = Q( d dt )u = P(s)Y (s) = Q(s)U(s) where Y := L (y) and U := L (u). The rational function Y (s)u 1 (s) = P 1 (s)q(s) =: H(s) is called transfer function. In the SISO case Y (s) U(s) = Q(s) P(s) =: h(s). ELEC 3035 (Part I, Lecture 2) State space and polynomial representations 11 / 19

9 State of the system a system B, Given a past trajectory of B, (...w p ( 2),w p ( 1)), and a future input u f = (u f (0),u f (1),...) find the future output y f of B, such that is a trajectory of B. w := (...,w p ( 2),w p ( 1),w f (0),w f (1),...) It turns out that for B = B i/o (p,q), it isn t necessary to know the whole (infinite) past w p in order to find y f! Suffices to know a finite dimensional, so called state, vector x(0) of B. ELEC 3035 (Part I, Lecture 2) State space and polynomial representations 12 / 19

10 Input/state/output (I/S/O) representation A finite dimensional LTI system B L w admits a representation B i/s/o (A,B,C,D) := {w := col(u,y) (R w ) N x (R n ) N, such that σx = Ax + Bu, y = Cx + Du }. (I/S/O repr) x an auxiliary variable called state n := dim(x) state dimension, R n state space A R n n, B R n m, C R p n, D R p m parameters of B m := dim(u) input dimension, p := dim(y) output dimension single input single output (SISO) systems dim(u) = dim(y) = 1 multi input multi output (MIMO) systems dim(u) 1, dim(y) 1 ELEC 3035 (Part I, Lecture 2) State space and polynomial representations 13 / 19

11 A state transition matrix, C output matrix, B input matrix D feedthrough matrix σx = Ax + Bu state equation y = Cx + Du output equation A shows how x(t + 1) depends on x(t) (state transition) B shows how u(t) influences x(t + 1) C shows how y(t) depends on x(t) D shows how u(t) influences y(t) (static I/O relation) Trivial extension: A, B, C, D functions of t leads to time-varying system ELEC 3035 (Part I, Lecture 2) State space and polynomial representations 14 / 19

12 Comparison between I/O and I/S/O representations (I/S/O repr) is first order in x and zeroth order in w (I/O repr) has no auxiliary variable and is for higher order in w If the system is single output, (I/S/O repr) is vector difference/differential equation (I/O repr) is a scalar difference/differential equation We will consider the problems of constructing I/S/O repr from an I/O one and vice verse, i.e., (P,Q) (A,B,C,D) and (A,B,C,D) (P,Q) ELEC 3035 (Part I, Lecture 2) State space and polynomial representations 15 / 19

13 Nonuniqueness of an I/S/O representation There are two sources of nonuniqueness of (I/S/O repr): 1. redundant states n := dim(x) bigger than necessary 2. nonuniqueness of A,B,C,D choice of state space basis minimal I/S/O representations dim(x) is as small as possible For any nonsingular matrix T R n n and A = T 1 AT, B = T 1 B, C = CT, D = D we have that B i/s/o (A,B,C,D) = B i/s/o ( A, B, C, D). ELEC 3035 (Part I, Lecture 2) State space and polynomial representations 16 / 19

14 Change of state space basis Consider an LTI system B = B i/s/o (A,B,C,D). For any (u,y) B, there is x, such that σx = Ax + Bu, y = Cx + Du. ( ) Let x = T 1 x, where T R n n is nonsingular, so that x = T x. Substituting in ( ) and multiplying the first equation by T, we obtain σ x = T 1 AT A x + T 1 B B u, y = CT C x + D u. D x = T x, with T nonsingular, means change of basis in R n (from I to T ). ELEC 3035 (Part I, Lecture 2) State space and polynomial representations 17 / 19

15 Nonuniqueness of an I/O representation There are two sources of nonuniqueness of (I/O repr): 1. redundant equations g := row dim(p) bigger than necessary 2. nonuniquencess of P, Q equivalence of equations minimal I/O representations row dim(p) is as small as possible In the single output case, P,Q are unique up do a scaling factor, i.e., P = αp, Q = αq, for α R we have that B i/o (P,Q) = B i/o ( P, Q). For multi output systems the nonuniqueness of P, Q is more essential. ELEC 3035 (Part I, Lecture 2) State space and polynomial representations 18 / 19

16 I/S/O transfer function The transfer function corresponding to a system B i/s/o (A,B,C,D) is H(s) = C(sI A) 1 B + D. With X := L (x), Y := L (y), U := L (u), we have The first equation implies σx = Ax + Bu = sx = AX + BU y = Cx + Du = Y = CX + DU (si A)X = BU = X = (si A) 1 BU. Substitute in the second equation to get Y = C(sI A) 1 BU + DU = C(sI A) 1 B + D H(s) U ELEC 3035 (Part I, Lecture 2) State space and polynomial representations 19 / 19